Conditions of solvability of the asynchronous spectrum control problem for linear periodic systems with upper left constant block of coefficient matrix
https://doi.org/10.29235/1561-8323-2026-70-2-95-101
Abstract
The focus of this study is a linear control system with a periodic matrix of coefficients and program control. The matrix under control is constant, the number of columns does not exceed the number of rows and its rank is less than the number of columns. It is assumed that the control is nontrivial periodic, and the module of its frequencies, i. e., the smallest additive group of real numbers, including all Fourier exponents of this control, is contained in the frequency module of the coefficient matrix. For the system under consideration, the problem of control of the asynchronous spectrum is posed: to construct such a control from an admissible set so that the system has strongly irregular periodic solutions. In this case, the period of the solution is incommensurate with the period of the system. Previously, the solution of the formulated problem was carried out for various cases of degeneracy of the average value of the coefficient matrix. In this work, a new approach is implemented that directly concerns the coefficient matrix itself. Under the assumption that its upper left block is stationary and the oscillating part of the upper right block has the maximum column rank, both necessary and also sufficient conditions for the solvability of the asynchronous spectrum control problem are obtained for the class of systems under consideration.
Keywords
periodic linear control systems,
highly irregular periodic solutions,
frequency spectrum of the solution,
asynchronous mode
Supporting agencies: The work was carried out at the Institute of Mathematics of the National Academy of Sciences of Belarus within the framework of the SPFI “Convergence– 2026”
About the Author
A. K. Demenchuk
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Demenchuk Aleksandr K. – D. Sc. (Physics and Mathematics), Professor, Chief Researcher
11, Surganov Str., 220072, Minsk
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